Properties

Label 6027.5470
Modulus $6027$
Conductor $287$
Order $120$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6027, base_ring=CyclotomicField(120))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,20,99]))
 
pari: [g,chi] = znchar(Mod(5470,6027))
 

Basic properties

Modulus: \(6027\)
Conductor: \(287\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(120\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{287}(17,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6027.dt

\(\chi_{6027}(19,\cdot)\) \(\chi_{6027}(313,\cdot)\) \(\chi_{6027}(766,\cdot)\) \(\chi_{6027}(913,\cdot)\) \(\chi_{6027}(1060,\cdot)\) \(\chi_{6027}(1195,\cdot)\) \(\chi_{6027}(1342,\cdot)\) \(\chi_{6027}(1489,\cdot)\) \(\chi_{6027}(1942,\cdot)\) \(\chi_{6027}(2236,\cdot)\) \(\chi_{6027}(2371,\cdot)\) \(\chi_{6027}(2518,\cdot)\) \(\chi_{6027}(2530,\cdot)\) \(\chi_{6027}(2677,\cdot)\) \(\chi_{6027}(2812,\cdot)\) \(\chi_{6027}(2959,\cdot)\) \(\chi_{6027}(2971,\cdot)\) \(\chi_{6027}(3265,\cdot)\) \(\chi_{6027}(3841,\cdot)\) \(\chi_{6027}(3988,\cdot)\) \(\chi_{6027}(4135,\cdot)\) \(\chi_{6027}(4147,\cdot)\) \(\chi_{6027}(4294,\cdot)\) \(\chi_{6027}(4441,\cdot)\) \(\chi_{6027}(5017,\cdot)\) \(\chi_{6027}(5311,\cdot)\) \(\chi_{6027}(5323,\cdot)\) \(\chi_{6027}(5470,\cdot)\) \(\chi_{6027}(5605,\cdot)\) \(\chi_{6027}(5752,\cdot)\) ...

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{120})$
Fixed field: Number field defined by a degree 120 polynomial (not computed)

Values on generators

\((4019,493,2794)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{33}{40}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 6027 }(5470, a) \) \(1\)\(1\)\(e\left(\frac{47}{60}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{59}{60}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{17}{120}\right)\)\(e\left(\frac{3}{40}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{47}{120}\right)\)\(e\left(\frac{31}{120}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6027 }(5470,a) \;\) at \(\;a = \) e.g. 2